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A combinatorial Hopf algebra is a graded connected Hopf algebra over a field k equipped with a character (multiplicative linear functional) Î¶ : H â†’ k. We show that the terminal object in the categoryâ€¦ (More)

We analyze the structure of the Malvenuto-Reutenauer Hopf algebra of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitiveâ€¦ (More)

We show that there are fewer than e 2 +3 4 2( k 2)nk positive solutions to a fewnomial system consisting of n polynomials in n variables having a total of n+k+1 distinct monomials. This isâ€¦ (More)

We show how to construct sparse polynomial systems that have non-trivial lower bounds on their numbers of real solutions. These are unmixed systems associated to certain polytopes. For the orderâ€¦ (More)

- Frank Sottile, Tom Marrie, +5 authors Anthony G. Gristina
- Critical care medicine
- 1986

Biomaterials are essential for life support and monitoring of critically ill patients, but their use increases the risk of nosocomial infection. Of the various plastics used for life support andâ€¦ (More)

- Frank Sottile, R Esum
- 1996

We establish the formula for multiplication by the class of a special Schubert variety in the integral cohomology ring of the ag manifold. This formula also describes the multiplication of a Schubertâ€¦ (More)

- Birkett Huber, Frank Sottile, Bernd Sturmfels
- J. Symb. Comput.
- 1998

We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge.â€¦ (More)

- Frank Sottile
- Experimental Mathematics
- 2000

Boris Shapiro and Michael Shapiro have a conjecture concerning the Schubert calculus and real enumerative geometry and which would give infinitely many families of zero-dimensional systems of realâ€¦ (More)

We show the equivalence of the Pieri formula for flag manifolds with certain identities among the structure constants for the Schubert basis of the polynomial ring. This gives new proofs of both theâ€¦ (More)

- Georgia Benkart, Frank Sottile, Jeffrey Stroomer
- J. Comb. Theory, Ser. A
- 1996

We deene and characterize switching, an operation that takes two tableaux sharing a common border and \moves them through each other" giving another such pair. Several authors, including James andâ€¦ (More)